Abstract
A Gizatullin surface is a normal affine surface V over
$ \mathbb{C} $
, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of
$ \mathbb{C}^{ * } $
-actions and
$ \mathbb{A}^{{\text{1}}} $
-fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with
$ \mathbb{C}_{{\text{ + }}} $
-actions on V considered up to a “speed change”.Non-Gizatullin surfaces are known to admit at most one
$ \mathbb{A}^{1} $
-fibration V → S up to an isomorphism of the base S. Moreover, an effective
$ \mathbb{C}^{ * } $
-action on them, if it does exist, is unique up to conjugation and inversion t
$ \mapsto $
t
−1 of
$ \mathbb{C}^{ * } $
. Obviously, uniqueness of
$ \mathbb{C}^{ * } $
-actions fails for affine toric surfaces. There is a further interesting family of nontoric Gizatullin surfaces, called the Danilov-Gizatullin surfaces, where there are in general several conjugacy classes of
$ \mathbb{C}^{ * } $
-actions and
$ \mathbb{A}^{{\text{1}}} $
-fibrations, see, e.g., [FKZ1].In the present paper we obtain a criterion as to when
$ \mathbb{A}^{{\text{1}}} $
-fibrations of Gizatullin surfaces are conjugate up to an automorphism of V and the base
$ S \cong \mathbb{A}^{{\text{1}}} $
. We exhibit as well large subclasses of Gizatullin
$ \mathbb{C}^{ * } $
-surfaces for which a
$ \mathbb{C}^{ * } $
-action is essentially unique and for which there are at most two conjugacy classes of
$ \mathbb{A}^{{\text{1}}} $
-fibrations over
$ \mathbb{A}^{{\text{1}}} $
.
